Skip to main content

Robust Connectivity of Graphs on Surfaces

  • Conference paper
  • First Online:
Extended Abstracts EuroComb 2021

Part of the book series: Trends in Mathematics ((RPCRMB,volume 14))


Let \(\varLambda (T)\) denote the set of leaves in a tree T. One natural problem is to look for a spanning tree T of a given graph G such that \(\varLambda (T)\) is as large as possible. Recently, a similar but stronger notion called the robust connectivity of a graph G was introduced, which is defined as the minimum value \(\frac{|R \cap \varLambda (T)|}{|R|}\) taken over all nonempty subsets \({R\subseteq V(G)}\), where \(T = T(R)\) is a spanning tree on G chosen to maximize \({|R \cap \varLambda (T)|}\). We prove a tight asymptotic bound of \(\varOmega (\gamma ^{-\frac{1}{r}})\) for the robust connectivity of r-connected graphs of Euler genus \(\gamma \). Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman.

T.M. was supported by a postdoctoral fellowship at the Simon Fraser University through NSERC grants R611450 and R611368. J.N. received support for foreign internships for the WDSMCS students in the framework of “Excellence initiative – research university” programme at MIM, University of Warsaw. L.S. was supported by NSERC grant R611368. The full version of the paper can be found on arXiv [4].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

    This parameter was formerly called “game connectivity” in [5]. However, we believe that the term “robust connectivity” better suits the properties of this parameter.


  1. Albertson, M.O., Berman, D.M.: A conjecture on planar graphs. Graph Theor. Relat. Top. 357, 1 (1979)

    Google Scholar 

  2. Bonamy, M., Kardoš, F., Kelly, T., Postle, L.: Fractional vertex-arboricity of planar graphs. arXiv:2009.12189 (2020)

  3. Borodin, O.V.: A proof of Grünbaum’s conjecture on the acyclic \(5\)-colorability of planar graphs (in Russian). Dokl. Akad. Nauk SSSR 231(1), 18–20 (1976)

    MathSciNet  Google Scholar 

  4. Bradshaw, P., Masařík, T., Novotná, J., Stacho, L.: Robust Connectivity of Graphs on Surfaces arXiv:2104.12030 (2021)

  5. Bradshaw, P., Masařík, T., Stacho, L.: Flexible list colorings in graphs with special degeneracy conditions. In: International Symposium on Algorithms and Computation, ISAAC 2020, pp. 31:1–31:15 (2020).

  6. Du, D.-Z., Wan, P.-J.: Connected Dominating Set: Theory and Applications. Springer, New York, (2013).

  7. Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., USA (1979)

    Google Scholar 

  8. Hosono, K.: Induced forests in trees and outerplanar graphs. Proc. Fac. Sci. Tokai Univ. (25), 27–29 (1990)

    Google Scholar 

  9. Kleitman, D.J., West, D.B.: Spanning trees with many leaves. SIAM J Discrete Math. 4(1), 99–106 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  10. Storer, J.A.: Constructing full spanning trees for cubic graphs. Inform. Process. Lett. 13(1), 8–11 (1981).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Tomáš Masařík .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bradshaw, P., Masařík, T., Novotná, J., Stacho, L. (2021). Robust Connectivity of Graphs on Surfaces. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham.

Download citation

Publish with us

Policies and ethics